The Stacks project

112.5.5 Cohomology

Papers discussing cohomology of sheaves on algebraic stacks.

• Olsson: Sheaves on Artin stacks [olsson_sheaves]

  This paper develops the theory of quasi-coherent and constructible sheaves proving basic cohomological properties. This paper corrects a mistake in [LM-B] in the functoriality of the lisse-étale site. The cotangent complex is constructed. In addition, the following theorems are proved: Grothendieck's Fundamental Theorem for proper morphisms, Grothendieck's Existence Theorem, Zariski's Connectedness Theorem and finiteness theorem for proper pushforwards of coherent and constructible sheaves.

• Behrend: Derived $l$-adic categories for algebraic stacks [behrend_derived]

  Proves the Lefschetz trace formula for algebraic stacks.

• Behrend: Cohomology of stacks [behrend_cohomology]

  Defines the de Rham cohomology for differentiable stacks and singular cohomology for topological stacks.

• Faltings: Finiteness of coherent cohomology for proper fppf stacks [faltings_finiteness]

  Proves coherence for direct images of coherent sheaves for proper morphisms.

• Abramovich, Corti, Vistoli: Twisted bundles and admissible covers [acv]

  The appendix contains the proper base change theorem for étale cohomology for tame Deligne-Mumford stacks.